Geometry and 2-Hilbert space for nonassociative magnetic translations

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We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non‐geometric fluxes. We survey approaches based on deformation quantization of twisted Poisson structures, symplectic realization of almost symplectic structures, and geometric quantization using 2‐Hilbert spaces of sections of suitable bundle gerbes. We compare and contrast these perspectives, describing their advantages and shortcomings in each case, and mention many open avenues for investigation.

“…A precise definition of nonassociative magnetic translations was constructed in [] on the 2‐Hilbert space

Γ (M, {scriptI}_{ρ})

in terms of the parallel transport functor

{scriptP}_{v} : Γ false(M, I_{ρ} false) ⟶ Γ false(M, I_{ρ} false)

which is defined on objects η as an infinitesimal version of the parallel transport operators and by the usual action of the translation group

T = {double-struckR}^{d}

on morphisms f via pullback

v^{*} false(f false)

\begin{matrix} true \\ right P_{v} {(η) false|}_{x} (a) & left {= η false|}_{x - v} (a) + \frac{1}{ℏ} \int_{▵^{1} (x; v)} ι_{a} ρ, \end{matrix}

\begin{matrix} true \\ right P_{v} (f) (x) & left = f false(x - v false), \end{matrix}

where

v \in T

x \in M

and

ι_{a}

denotes contraction with the vector

a \in {double-struckR}^{d}

. This definition can be understood by transgressing the gerbe

I_{ρ}

to a line bundle with connection over the loop space

L M

of M , and defining parallel transport over

L M

in the usual...…”

Section: Perspective Iii: Higher Geometric Quantizationmentioning

confidence: 99%

H = 0

case,

ω_{u, v, w}

define a three‐cocycle on

R^{d}

with values in

C^{\infty} false(M, normalU (1) false)

.…”

Section: Perspective Iii: Higher Geometric Quantizationmentioning

confidence: 99%

“…On the other hand, the pairs

(χ_{v, w}, ω_{u, v, w})

define a higher weak two‐cocycle on

R^{d}

with values in the Hilb‐algebra category

HVbdl (M)

. These constructions were collected together in [] to give the following central result. Theorem The pairs

({scriptP}_{v}, {normalΠ}_{v, w})

define a weak projective 2‐representation of the translation group

R^{d}

on the

HVbdl (M)

‐module category

Γ (M, {scriptI}_{ρ})

, the 2‐Hilbert space of sections of the bundle gerbe

I_{ρ}

on M .…”

Section: Perspective Iii: Higher Geometric Quantizationmentioning

confidence: 99%

Γ (M, {scriptI}_{ρ})

f \in C^{\infty} false(scriptM false)

, regarded as objects in the functor category

[M ⇉ M, C ⇉ C]

between the discrete categories based on the sets

scriptM

and

double-struckC

, to objects

O_{f}

in the functor category

[Γ (M, G), Γ (M, G)]

.…”

Section: Perspective Iii: Higher Geometric Quantizationmentioning

confidence: 99%

“…Here we are glossing over many technical functional analytic details: The domain of the magnetic Weyl transform is smaller than the smooth functions

C^{\infty} false(scriptM false)

and the Weyl operators

W (x, p)

only act on a Schwartz subspace of

scriptH

in a suitable way; see e.g. [] for the precise treatment. These details are not important for the discussion that follows.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Quantization of Magnetic Poisson Structures

Szabo

2019

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Para‐Hermitian Geometry, Dualities and Generalized Flux Backgrounds

Marotta

Szabo

2018

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We survey physical models which capture the main concepts of double field theory on para-Hermitian manifolds. We show that the geometric theory of Lagrangian and Hamiltonian dynamical systems is an instance of para-Kähler geometry which extends to a natural example of a Born geometry. The corresponding phase space geometry belongs to the family of natural almost para-Kähler structures which we construct explicitly as deformations of the canonical para-Kähler structure by non-linear connections. We extend this framework to a class of non-Lagrangian dynamical systems which naturally encodes the notion of fluxes in para-Hermitian geometry. In this case we describe the emergence of fluxes in terms of weak integrability defined by the D-bracket, and we extend the construction to arbitrary cotangent bundles where we reproduce the standard generalized fluxes of double field theory. We also describe the para-Hermitian geometry of Drinfel'd doubles, which gives an explicit illustration of the interplay between fluxes, D-brackets and different polarizations. The left-invariant para-Hermitian structure on a Drinfel'd double in a Manin triple polarization descends to a doubled twisted torus, which we use to illustrate how changes of polarizations give rise to different fluxes and string backgrounds in para-Hermitian geometry.1 An alternative mathematical approach to double field theory based on graded symplectic geometry is discussed in [21][22][23].

Global Double Field Theory is Higher Kaluza‐Klein Theory

Alfonsi

2020

Kaluza-Klein Theory states that a metric on the total space of a principal bundle P → M, if it is invariant under the principal action of P, naturally reduces to a metric together with a gauge field on the base manifold M. We propose a generalization of this Kaluza-Klein principle to higher principal bundles and higher gauge fields. For the particular case of the abelian gerbe of Kalb-Ramond field, this Higher Kaluza-Klein geometry provides a natural global formulation for Double Field Theory (DFT). In this framework the doubled space is the total space of a higher principal bundle and the invariance under its higher principal action is exactly a global formulation of the familiar strong constraint. The patching problem of DFT is naturally solved by gluing the doubled space with a higher group of symmetries in a higher category. Locally we recover the familiar picture of an ordinary para-Hermitian manifold equipped with Born geometry. Infinitesimally we recover the familiar picture of a higher Courant algebroid twisted by a gerbe (also known as Extended Riemannian Geometry). As first application we show that on a torus-compactified spacetime the Higher Kaluza-Klein reduction gives automatically rise to abelian T-duality, while on a general principal bundle it gives rise to non-abelian T-duality. As final application we define a natural notion of Higher Kaluza-Klein monopole by directly generalizing the ordinary Gross-Perry one. Then we show that under Higher Kaluza-Klein reduction, this monopole is exactly the NS5-brane on a 10d spacetime. If, instead, we smear it along a compactified direction we recover the usual DFT monopole on a 9d spacetime.